Blow-up for quasilinear heat equations with critical Fujita's exponents
Blow-up for quasilinear heat equations with critical Fujita's exponents
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DOI：
10.1017/s0308210500028766
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发表时间：
1994
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影响因子：
--
通讯作者：
V. Galaktionov
中科院分区：
文献类型：
--
作者：
V. Galaktionov
考虑拟线性热方程的Cauchy问题，其中σ > 0是一个固定常数，临界指数在源项β = βc = σ + 1 + 2/N.众所周知，如果β ∈（1，βc），则任何非负弱解u（x，t）<$0在有限时间内爆破.对于σ = 0的半线性热方程（HE），H. Fujita [4].本文证明了u 0在临界情况β = σ + 1 + 2/N且σ > 0时爆炸。对于σ > 0，临界指数β = σ + 1 +（σ + 2）/N的梯度扩散方程，也有类似的结果。
We consider the Cauchy problem for the quasilinear heat equation where σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4]. In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivity with σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.